Harmonic Analysis (Summer term 2020) - December 13, 2019
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Harmonic Analysis (Summer term 2020)
December 13, 2019
Lectures: Wed 16.45-18.15 (in F 513) and Thu 8.00-9.30 (in F 315), starting
on April 15, 2020
Exercises: Thu 9.45-11.15 (in F 315)
Analysis in general tends to revolve around the study of general classes of
functions and operators. Real-variable harmonic analysis focuses in particular
on the relation between qualitative properties (such as measurability, bounded-
ness, differentiability, analyticity, integrability, decay at infinity, convergence,
etc.) and their quantification (i.e., what is the smallest upper bound on a func-
tion, how often is it differentiable, what is its Lp norm, what is the convergence
rate of a sequence, etc.). It is then natural to ask how quantitative properties of
such functions change when one applies various (often quite explicit) operators.
It turns out that quantitative estimates, such as Lp estimates on such operators,
provide an important route to establish qualitative results and in fact there are
a number of principles (such as the uniform boundedness principle or Stein’s
maximal principle [29]) which assert that this is the only route, in the sense
that a quantitative result must exist in order for the qualitative result to be
true.
Many arguments in harmonic analysis will, at some point, involve a com-
binatorial statement about certain types of geometric objects such as cubes,
balls, or tubes. One such useful statement is the Vitali covering lemma which
asserts that given a collection of balls B1 , ..., Bk in Euclidean space, then there
exists a subcollection of balls Bi1 , ..., Bim which are disjoint but contain a sig-
nificant
Sk fraction S
of the volume covered by the original balls, in the sense that
m
| i=1 Bk | ≤ ad | j=1 Bij | for some d-dependent constant ad .
One feature of harmonic analysis methods is that they tend to be local rather
than global. For instance, it is quite common to analyze a function f by applying
cutoff functions in either the spatial or the frequency variables to decompose f
into a number of somewhat localized pieces. One then estimates each of these
pieces separately and “glues” the estimates back together at the end. One rea-
son for this “divide and conquer” strategy is that generic functions tend to have
infinitely many degrees of freedom (f may for instance be very smooth but
slowly decaying at one place whereas at other places f may be highly singular
or oscillating very quickly) and it would be quite difficult to treat all of these
features at once. A well chosen decomposition can isolate these features from
1each other, so that each component only has one salient feature that could cause
difficulty. In reassembling the estimates from the individual components, one
can use rather crude tools such as the triangle inequality, or more refined tools,
such as ones relying on (almost) orthogonality. The main drawback of decom-
position methods is however that one generally does not obtain the optimal
constants.
Another basic theme of harmonic analysis is the attempt to quantify the
elusive phenomenon of oscillation. Intuitively, if an expression oscillates wildly
in phase, then its average value should be relatively small in amplitude. This
leads to the principle of stationary phase and the Heisenberg uncertainty prin-
ciple which relates the decay and smoothness of a function to the smoothness
and the decay of its Fourier transform. The development of a robust theory
for oscillatory integrals is also one main ingredient to understand the interplay
between Lp estimates of certain Fourier multipliers (such as the Bochner–Riesz
means) and geometric properties of certain (smooth) manifolds, such as the
Fourier transform of the associated surface measure and Lp estimates for it.
1. Interpolation theorems (Marcinkiewicz (in particular restricted weak-type
formulation, see, e.g., Tao [34, Lecture 2]), Riesz–Thorin, Stein [33])
2. Covering lemmas (Vitali, Whitney, Calderón–Zygmund) as well as Calderón–
Zygmund decomposition following Stein [30, Chapter 1], see also Grafakos
[17, Proposition 2.1.20, Theorem 4.3.1], and Guzmán [11]
3. Maximal functions following Stein [30, 32]
(a) Hardy–Littlewood maximal function and Lebesgue differentiation the-
orem following Stein [30, Chapter 1], [32, Chapter I, Section §3]
(b) Maximal functions and Lebesgue differentiation for more general sets
(instead of balls), see Guzmán [11], in particular Cordoba–Fefferman
[9]
(c) Hardy–Littlewood p maximal function (see Blunck–Kunstmann [3,
1])
(d) Relation to convergence almost everywhere, first glance at Bochner–
Riesz summability (FAP1 and FAP2 in [22]. In this regard, see also
the “ergodic Hopf–Dunford–Schwartz” theorem [31, p. 48] respec-
tively Dunford–Schwartz [12] (Section XIII.6: Lemma 7 (p. 676),
Theorem 8 (p. 678); Section XIII.8: Lemma 6 (p. 690) Theorem 7
(p. 693); Section XIII.9: Exercise 3 (p. 717)))
4. Singular integrals following Stein [30, Chapter II], [32, Chapter I, Section
§5]. In particular, Hilbert transform and its application to partial sums
operators [30, Chapter IV, Section §4] and second glance at Bochner–Riesz
(box multiplier versus disc multiplier, see also Fefferman [15, 16])
5. Riesz transforms and Poisson integrals following Stein [30, Chapter III]
and [31, Section §4.4]
26. A primer on the Fourier transform (Wolff [38, Chapters 1-5]) and Mikhlin–
Hörmander multiplier theorem for Fourier multipliers [32, Chapter VI,
Section §4.4] (see also Sogge [28, Theorem 0.2.6]) and square function es-
timates / Littlewood–Paley inequalities [23] (rough version with dyadic
cubes as in Stein [30, pp. 103-108] or Duoandikoetxea [13, Theorem 8.4]
or with dyadic annuli as in Tao [37, Lecture 2, Theorem 1] or arbitrary
intervals (when all intervals have the same length, the result is sharp, see
Carleson [8], Córdoba [10], and Rubio de Francia [25]) as in Rubio de
Francia [26]; smooth version using bump functions or heat kernels as in
Killip et al [21, Theorem 4.3] or [20, Theorem 5.3]). Generalization to gen-
eral self-adjoint operators (such as Schrödinger operators) instead of mere
Fourier multipliers: spectral multiplier theorems, Bernstein estimates, and
Littlewood–Paley inequalities and their application in nonlinear PDEs
7. Introduction to pseudodifferential operators following Stein [32, Chapter
VI] (see also Martinez [24] (Chapter 2, in particular from Section 2.5 on))
(a) Symbolic calculus, composition [32, Chapter VI, Section §3] and [24,
Sections 2.6, 2.7]
(b) L2 boundedness, Calderón–Vaillancourt theorem [32, Chapter VI,
Section §2] and [24, Section 2.8]
(c) Singular integral representation, bounds on integral kernels [32, Chap-
ter VI, Section §4.1-4.3]
(d) L2 boundedness of translation invariant Calderón–Zygmund opera-
tors [32, Chapter VI, Section §4.5]
(e) Estimates in Lp , Sobolev, and Lipschitz spaces [32, Chapter VI, Sec-
tion §5]
8. More on (spectral) multiplier theorems. See in particular Hebisch [18],
Duong–Ouhabaz–Sikora [14], Blunck–Kunstmann [3], and Blunck [2]
9. Almost orthogonality following Stein [32, Chapter VII]
(a) Exotic and forbidden symbols, failure of L2 boundedness for symbols
0
in S1,1 [32, Chapter VII, Section §1]
(b) Cotlar–Stein lemma [32, Chapter VII, Section §2.1-2.3] and general-
ization to Schatten classes, see Carbery [7]
0
(c) Consequences of Cotlar–Stein for symbols in Sρ,ρ (with 0 ≤ ρ < 1)
[32, Chapter VII, Section §2.4-2.5]
(d) L2 theory for Calderón–Zygmund operators [32, Chapter VII, Section
§3]
(e) More on the Cauchy integral [32, Chapter VII, Section §4]
10. Uncertainty principle following Wolff [38, Chapter 5]
11. Oscillatory integrals following Stein [32, Chapter XIII, IX]
3(a) Oscillatory integrals of the first kind, stationary phase [32, Chapter
XIII, Section §1-2]
(b) Fourier transform of surface measures [32, Chapter XIII, Section
§3] and application to the lattice counting problem (improvement
of Weyl’s law for −∆ on Td ) (Sogge [27, pp. 83-85])
(c) Introduction to Fourier restriction [32, Chapter XIII, Section §4]
(d) Oscillatory integrals of the second kind, Carleson–Sjölin and Hörman-
der integral operators [32, Chapter IX, Section §1], see also Bourgain
[4]
(e) Relation to Fourier restriction and Bochner–Riesz summability [32,
Chapter IX, Section §2], Sogge [28, Sections 2.2-2.3], Tao [34, 35, 36]
12. Decoupling inequalities [5, 6, 19]
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